ABCF->ab-angle a

Average Error: 52.5 → 35.0

Time: 53.3s

Precision: binary64

Cost: 40520

Math

\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{A + \left(C + t_0\right)}\\ t_2 := -\sqrt{F}\\ t_3 := \frac{\sqrt{2}}{B}\\ t_4 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_5 := -4 \cdot \left(A \cdot C\right)\\ t_6 := A + \mathsf{hypot}\left(B, A\right)\\ \mathbf{if}\;B \leq -1.45 \cdot 10^{+112}:\\ \;\;\;\;t_3 \cdot \left(\sqrt{t_6} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_4} \cdot \left(t_1 \cdot t_2\right)}{t_4}\\ \mathbf{elif}\;B \leq -1.65 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq -1.05 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + t_0\right)\right)} \cdot \left(-\sqrt{F \cdot \left(A \cdot \left(C \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{+32}:\\ \;\;\;\;\frac{-t_1 \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_5\right)\right)}}{B \cdot B + t_5}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{t_0 + \left(A + C\right)} \cdot \frac{B}{\frac{t_4}{\sqrt{2} \cdot t_2}}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(-\sqrt{t_6 \cdot F}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
     (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
  (- (pow B 2.0) (* (* 4.0 A) C))))

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (hypot B (- A C)))
        (t_1 (sqrt (+ A (+ C t_0))))
        (t_2 (- (sqrt F)))
        (t_3 (/ (sqrt 2.0) B))
        (t_4 (fma B B (* C (* A -4.0))))
        (t_5 (* -4.0 (* A C)))
        (t_6 (+ A (hypot B A))))
   (if (<= B -1.45e+112)
     (* t_3 (* (sqrt t_6) (sqrt F)))
     (if (<= B -6e-88)
       (/ (* (sqrt (* 2.0 t_4)) (* t_1 t_2)) t_4)
       (if (<= B -1.65e-190)
         (sqrt (/ (- F) C))
         (if (<= B -1.05e-291)
           (/
            (*
             (sqrt (* 2.0 (+ C (+ A t_0))))
             (- (sqrt (* F (* A (* C -4.0))))))
            (fma -4.0 (* A C) (* B B)))
           (if (<= B 4e-206)
             (sqrt (- (/ F A)))
             (if (<= B 2.3e+32)
               (/
                (- (* t_1 (sqrt (* 2.0 (* F (fma B B t_5))))))
                (+ (* B B) t_5))
               (if (<= B 3.6e+157)
                 (* (sqrt (+ t_0 (+ A C))) (/ B (/ t_4 (* (sqrt 2.0) t_2))))
                 (* t_3 (- (sqrt (* t_6 F)))))))))))))

C

double code(double A, double B, double C, double F) {
	return -sqrt(((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / (pow(B, 2.0) - ((4.0 * A) * C));
}

↓

double code(double A, double B, double C, double F) {
	double t_0 = hypot(B, (A - C));
	double t_1 = sqrt((A + (C + t_0)));
	double t_2 = -sqrt(F);
	double t_3 = sqrt(2.0) / B;
	double t_4 = fma(B, B, (C * (A * -4.0)));
	double t_5 = -4.0 * (A * C);
	double t_6 = A + hypot(B, A);
	double tmp;
	if (B <= -1.45e+112) {
		tmp = t_3 * (sqrt(t_6) * sqrt(F));
	} else if (B <= -6e-88) {
		tmp = (sqrt((2.0 * t_4)) * (t_1 * t_2)) / t_4;
	} else if (B <= -1.65e-190) {
		tmp = sqrt((-F / C));
	} else if (B <= -1.05e-291) {
		tmp = (sqrt((2.0 * (C + (A + t_0)))) * -sqrt((F * (A * (C * -4.0))))) / fma(-4.0, (A * C), (B * B));
	} else if (B <= 4e-206) {
		tmp = sqrt(-(F / A));
	} else if (B <= 2.3e+32) {
		tmp = -(t_1 * sqrt((2.0 * (F * fma(B, B, t_5))))) / ((B * B) + t_5);
	} else if (B <= 3.6e+157) {
		tmp = sqrt((t_0 + (A + C))) * (B / (t_4 / (sqrt(2.0) * t_2)));
	} else {
		tmp = t_3 * -sqrt((t_6 * F));
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)))
end

↓

function code(A, B, C, F)
	t_0 = hypot(B, Float64(A - C))
	t_1 = sqrt(Float64(A + Float64(C + t_0)))
	t_2 = Float64(-sqrt(F))
	t_3 = Float64(sqrt(2.0) / B)
	t_4 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_5 = Float64(-4.0 * Float64(A * C))
	t_6 = Float64(A + hypot(B, A))
	tmp = 0.0
	if (B <= -1.45e+112)
		tmp = Float64(t_3 * Float64(sqrt(t_6) * sqrt(F)));
	elseif (B <= -6e-88)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_4)) * Float64(t_1 * t_2)) / t_4);
	elseif (B <= -1.65e-190)
		tmp = sqrt(Float64(Float64(-F) / C));
	elseif (B <= -1.05e-291)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(C + Float64(A + t_0)))) * Float64(-sqrt(Float64(F * Float64(A * Float64(C * -4.0)))))) / fma(-4.0, Float64(A * C), Float64(B * B)));
	elseif (B <= 4e-206)
		tmp = sqrt(Float64(-Float64(F / A)));
	elseif (B <= 2.3e+32)
		tmp = Float64(Float64(-Float64(t_1 * sqrt(Float64(2.0 * Float64(F * fma(B, B, t_5)))))) / Float64(Float64(B * B) + t_5));
	elseif (B <= 3.6e+157)
		tmp = Float64(sqrt(Float64(t_0 + Float64(A + C))) * Float64(B / Float64(t_4 / Float64(sqrt(2.0) * t_2))));
	else
		tmp = Float64(t_3 * Float64(-sqrt(Float64(t_6 * F))));
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := N[((-N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[F], $MachinePrecision])}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$4 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.45e+112], N[(t$95$3 * N[(N[Sqrt[t$95$6], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -6e-88], N[(N[(N[Sqrt[N[(2.0 * t$95$4), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[B, -1.65e-190], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], If[LessEqual[B, -1.05e-291], N[(N[(N[Sqrt[N[(2.0 * N[(C + N[(A + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 4e-206], N[Sqrt[(-N[(F / A), $MachinePrecision])], $MachinePrecision], If[LessEqual[B, 2.3e+32], N[((-N[(t$95$1 * N[Sqrt[N[(2.0 * N[(F * N[(B * B + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 3.6e+157], N[(N[Sqrt[N[(t$95$0 + N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B / N[(t$95$4 / N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * (-N[Sqrt[N[(t$95$6 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if B < -1.4500000000000001e112

   1. Initial program 61.4

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 61.0

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]
      Proof

      [Start] 61.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 63.8

      \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)\right)}{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}}} \]

   4. Simplified 63.6

      \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\left(2 \cdot F\right) \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}} \]
      Proof

      [Start] 63.8	\[ \sqrt{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)\right)}{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}} \]
      associate-/l* [=>] 63.6	\[ \sqrt{\color{blue}{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}}} \]
      *-commutative [=>] 63.6	\[ \sqrt{\frac{\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}} \]
      *-commutative [=>] 63.6	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)\right)}^{2}}{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}} \]
      *-commutative [=>] 63.6	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\color{blue}{\left(2 \cdot F\right)} \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}} \]
      +-commutative [=>] 63.6	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\left(2 \cdot F\right) \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}} \]
      +-commutative [=>] 63.6	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\left(2 \cdot F\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)}}} \]

   5. Taylor expanded in C around 0 58.8

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]

   6. Simplified 58.8

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F}} \]
      Proof

      [Start] 58.8	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \]
      unpow2 [=>] 58.8	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right) \cdot F} \]
      unpow2 [=>] 58.8	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right) \cdot F} \]

   7. Applied egg-rr 13.8

      \[\leadsto \frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]

   if -1.4500000000000001e112 < B < -5.9999999999999999e-88

   1. Initial program 44.8

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 40.6

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
      Proof

      [Start] 44.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 37.3

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Simplified 37.0

      \[\leadsto \frac{-\color{blue}{\sqrt{F \cdot \left(C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      Proof

      [Start] 37.3	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 37.3	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+r+ [=>] 37.0	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(\mathsf{hypot}\left(B, A - C\right) + C\right) + A\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [<=] 37.0	\[ \frac{-\sqrt{F \cdot \left(\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right)} + A\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+l+ [=>] 37.0	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Applied egg-rr 32.9

      \[\leadsto \frac{-\color{blue}{\left(\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \sqrt{F}\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   6. Simplified 32.5

      \[\leadsto \frac{-\color{blue}{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F}\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      Proof

      [Start] 32.9	\[ \frac{-\left(\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+r+ [=>] 32.5	\[ \frac{-\left(\sqrt{\color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + C\right) + A}} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [=>] 32.5	\[ \frac{-\left(\sqrt{\color{blue}{A + \left(\mathsf{hypot}\left(B, A - C\right) + C\right)}} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [=>] 32.5	\[ \frac{-\left(\sqrt{A + \color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if -5.9999999999999999e-88 < B < -1.65000000000000009e-190

   1. Initial program 50.2

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 44.4

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]
      Proof

      [Start] 50.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 58.9

      \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)\right)}{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}}} \]

   4. Simplified 58.8

      \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\left(2 \cdot F\right) \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}} \]
      Proof

      [Start] 58.9	\[ \sqrt{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)\right)}{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}} \]
      associate-/l* [=>] 58.8	\[ \sqrt{\color{blue}{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}}} \]
      *-commutative [=>] 58.8	\[ \sqrt{\frac{\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}} \]
      *-commutative [=>] 58.8	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)\right)}^{2}}{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}} \]
      *-commutative [=>] 58.8	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\color{blue}{\left(2 \cdot F\right)} \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}} \]
      +-commutative [=>] 58.8	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\left(2 \cdot F\right) \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}} \]
      +-commutative [=>] 58.8	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\left(2 \cdot F\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)}}} \]

   5. Taylor expanded in C around -inf 53.5

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{F}{C}}} \]

   6. Simplified 53.5

      \[\leadsto \sqrt{\color{blue}{-\frac{F}{C}}} \]
      Proof

      [Start] 53.5	\[ \sqrt{-1 \cdot \frac{F}{C}} \]
      mul-1-neg [=>] 53.5	\[ \sqrt{\color{blue}{-\frac{F}{C}}} \]

   if -1.65000000000000009e-190 < B < -1.05e-291

   1. Initial program 50.7

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 45.5

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]
      Proof

      [Start] 50.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around inf 47.3

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   4. Simplified 45.5

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(F \cdot \left(-4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      Proof

      [Start] 47.3	\[ \frac{-\sqrt{\left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-*r* [=>] 45.5	\[ \frac{-\sqrt{\left(-4 \cdot \color{blue}{\left(\left(A \cdot C\right) \cdot F\right)}\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-*l* [<=] 45.5	\[ \frac{-\sqrt{\color{blue}{\left(\left(-4 \cdot \left(A \cdot C\right)\right) \cdot F\right)} \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [<=] 45.5	\[ \frac{-\sqrt{\color{blue}{\left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 45.5	\[ \frac{-\sqrt{\left(F \cdot \left(-4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   5. Applied egg-rr 42.3

      \[\leadsto \frac{-\color{blue}{\sqrt{F \cdot \left(A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   if -1.05e-291 < B < 4.00000000000000011e-206

   1. Initial program 52.6

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 46.7

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]
      Proof

      [Start] 52.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 58.1

      \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)\right)}{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}}} \]

   4. Simplified 58.0

      \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\left(2 \cdot F\right) \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}} \]
      Proof

      [Start] 58.1	\[ \sqrt{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)\right)}{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}} \]
      associate-/l* [=>] 58.0	\[ \sqrt{\color{blue}{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}}} \]
      *-commutative [=>] 58.0	\[ \sqrt{\frac{\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{2}}{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}} \]
      *-commutative [=>] 58.0	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)\right)}^{2}}{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}} \]
      *-commutative [=>] 58.0	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\color{blue}{\left(2 \cdot F\right)} \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}} \]
      +-commutative [=>] 58.0	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\left(2 \cdot F\right) \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}} \]
      +-commutative [=>] 58.0	\[ \sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\frac{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}}{\left(2 \cdot F\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)}}} \]

   5. Taylor expanded in C around inf 49.3

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]

   6. Simplified 49.3

      \[\leadsto \sqrt{\color{blue}{-\frac{F}{A}}} \]
      Proof

      [Start] 49.3	\[ \sqrt{-1 \cdot \frac{F}{A}} \]
      mul-1-neg [=>] 49.3	\[ \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 4.00000000000000011e-206 < B < 2.3e32

   1. Initial program 47.9

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 47.9

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      Proof

      [Start] 47.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 37.9

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Simplified 37.9

      \[\leadsto \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof

      [Start] 37.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 37.9	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 37.9	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 2.3e32 < B < 3.60000000000000024e157

   1. Initial program 49.0

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 49.0

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      Proof

      [Start] 49.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 35.8

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Simplified 35.8

      \[\leadsto \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof

      [Start] 35.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 35.8	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 35.8	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around inf 29.3

      \[\leadsto \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot B\right) \cdot \sqrt{F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Applied egg-rr 26.3

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}}{1} \cdot \frac{\left(B \cdot \sqrt{2}\right) \cdot \left(-\sqrt{F}\right)}{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}} \]

   7. Simplified 33.1

      \[\leadsto \color{blue}{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \frac{B}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\sqrt{2} \cdot \left(-\sqrt{F}\right)}}} \]
      Proof

      [Start] 26.3	\[ \frac{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}}{1} \cdot \frac{\left(B \cdot \sqrt{2}\right) \cdot \left(-\sqrt{F}\right)}{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)} \]
      /-rgt-identity [=>] 26.3	\[ \color{blue}{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}} \cdot \frac{\left(B \cdot \sqrt{2}\right) \cdot \left(-\sqrt{F}\right)}{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)} \]
      +-commutative [=>] 26.3	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \color{blue}{\left(C + A\right)}} \cdot \frac{\left(B \cdot \sqrt{2}\right) \cdot \left(-\sqrt{F}\right)}{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)} \]
      associate-*l* [=>] 26.4	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \frac{\color{blue}{B \cdot \left(\sqrt{2} \cdot \left(-\sqrt{F}\right)\right)}}{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)} \]
      associate-/l* [=>] 33.1	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \color{blue}{\frac{B}{\frac{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}{\sqrt{2} \cdot \left(-\sqrt{F}\right)}}} \]
      *-commutative [=>] 33.1	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \frac{B}{\frac{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)}{\sqrt{2} \cdot \left(-\sqrt{F}\right)}} \]
      *-commutative [=>] 33.1	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \frac{B}{\frac{\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(A \cdot -4\right)}\right)}{\sqrt{2} \cdot \left(-\sqrt{F}\right)}} \]

   if 3.60000000000000024e157 < B

   1. Initial program 64.0

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
      Proof

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 64.0

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Simplified 64.0

      \[\leadsto \frac{-\color{blue}{\sqrt{F \cdot \left(C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      Proof

      [Start] 64.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 64.0	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+r+ [=>] 64.0	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(\mathsf{hypot}\left(B, A - C\right) + C\right) + A\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [<=] 64.0	\[ \frac{-\sqrt{F \cdot \left(\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right)} + A\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+l+ [=>] 64.0	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Applied egg-rr 64.0

      \[\leadsto \frac{-\color{blue}{\left(\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \sqrt{F}\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   6. Simplified 64.0

      \[\leadsto \frac{-\color{blue}{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F}\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      Proof

      [Start] 64.0	\[ \frac{-\left(\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+r+ [=>] 64.0	\[ \frac{-\left(\sqrt{\color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + C\right) + A}} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [=>] 64.0	\[ \frac{-\left(\sqrt{\color{blue}{A + \left(\mathsf{hypot}\left(B, A - C\right) + C\right)}} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [=>] 64.0	\[ \frac{-\left(\sqrt{A + \color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   7. Taylor expanded in C around 0 64.0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]

   8. Simplified 33.0

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)} \]
      Proof

      [Start] 64.0	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      mul-1-neg [=>] 64.0	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]
      distribute-rgt-neg-in [=>] 64.0	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
      *-commutative [=>] 64.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{B}^{2} + {A}^{2}}\right)}}\right) \]
      unpow2 [=>] 64.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A + \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)}\right) \]
      unpow2 [=>] 64.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A + \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)}\right) \]
      hypot-def [=>] 33.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A + \color{blue}{\mathsf{hypot}\left(B, A\right)}\right)}\right) \]

3. Recombined 8 regimes into one program.

4. Final simplification 35.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.45 \cdot 10^{+112}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{F}\right)\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -1.65 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq -1.05 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{F \cdot \left(A \cdot \left(C \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{+32}:\\ \;\;\;\;\frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)} \cdot \frac{B}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\sqrt{2} \cdot \left(-\sqrt{F}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\left(A + \mathsf{hypot}\left(B, A\right)\right) \cdot F}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	34.1
Cost	34132

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \frac{-\sqrt{A + \left(C + t_2\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)}}{B \cdot B + t_0}\\ t_4 := A + \mathsf{hypot}\left(B, A\right)\\ \mathbf{if}\;B \leq -2.9 \cdot 10^{+107}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{t_4} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-291}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{+33}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{t_2 + \left(A + C\right)} \cdot \frac{B}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\sqrt{2} \cdot \left(-\sqrt{F}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{t_4 \cdot F}\right)\\ \end{array} \]

Alternative 2
Error	34.1
Cost	28112

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)}}{B \cdot B + t_0}\\ t_2 := \frac{\sqrt{2}}{B}\\ t_3 := A + \mathsf{hypot}\left(B, A\right)\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{+108}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{t_3} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -6.6 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 2.75 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{t_3 \cdot F}\right)\\ \end{array} \]

Alternative 3
Error	38.6
Cost	27728

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := A + \mathsf{hypot}\left(B, A\right)\\ \mathbf{if}\;B \leq -2.5 \cdot 10^{-87}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{t_1} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.02 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{F} \cdot \left(\sqrt{2} \cdot \left(-B\right)\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{t_1 \cdot F}\right)\\ \end{array} \]

Alternative 4
Error	40.1
Cost	26372

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{-87}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-t_0\right)\\ \end{array} \]

Alternative 5
Error	40.2
Cost	21192

\[\begin{array}{l} \mathbf{if}\;B \leq -1.55 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{F} \cdot \frac{\sqrt{2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(B \cdot \left(B \cdot F\right)\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 6
Error	39.9
Cost	20564

\[\begin{array}{l} t_0 := \sqrt{F} \cdot \frac{\sqrt{2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{B}\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -9.5 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-67}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -5.8 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 7
Error	43.1
Cost	20236

\[\begin{array}{l} t_0 := \sqrt{F} \cdot \frac{\sqrt{2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{B}\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -1 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.28 \cdot 10^{-67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	45.1
Cost	19908

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.35 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{A - B} \cdot \left(\sqrt{2} \cdot \frac{\sqrt{F}}{B}\right)\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -5.2 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	43.8
Cost	19908

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.6 \cdot 10^{+105}:\\ \;\;\;\;\frac{\sqrt{F}}{\frac{\frac{B}{\sqrt{2}}}{\sqrt{A - B}}}\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -6.2 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	46.5
Cost	15440

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.35 \cdot 10^{+105}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{B \cdot \left(-F\right)}\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -2.8 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	48.7
Cost	15056

\[\begin{array}{l} t_0 := \frac{-\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{B \cdot \left(-F\right)}\\ \mathbf{elif}\;B \leq -6.2 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	49.7
Cost	13444

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -7.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{B \cdot \left(-F\right)}\\ \mathbf{elif}\;B \leq -4.8 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 13
Error	49.8
Cost	8716

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -5.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(A - B\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -1.05 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 14
Error	51.1
Cost	7108

\[\begin{array}{l} \mathbf{if}\;B \leq -3.15 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(A - B\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \end{array} \]

Alternative 15
Error	54.2
Cost	7053

\[\begin{array}{l} \mathbf{if}\;A \leq -7000000000 \lor \neg \left(A \leq 2.8 \cdot 10^{-246}\right) \land A \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 16
Error	56.9
Cost	6656

\[\sqrt{-\frac{F}{A}} \]

Alternative 17
Error	63.3
Cost	6592

\[\sqrt{\frac{F}{A}} \]

Reproduce

herbie shell --seed 2023016 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))